Back in school again Maxwell (PEPE) plays the fool again. / Teacher gets annoyed.
Wishing to Avoid and Unpleasant / Sce, e, e, Jan, / She tells Max to stay When the class has gone away, / So I waits behind / Writing fifty times "I Must Not Be / So, or, or, or ...
MAXWELL'S SILVER HAMMER (The Beatles, Abbey Road)
Well, let me tell you that now I was reading the Hall & Knight, a legendary algebra book perhaps because it is too demanding on the reader. The issue: finite differences as a way to get the nth term of a sequence, and the sum of the series to n terms. And I wondered why such methods as are known in elementary school mathematics? I think the answer may be in the school mathematics culture: if nobody uses them, the predictable result is that public knowledge tends to die out, is the strength of the performance measures (standards) imposed by the community and their environment.
But let us put aside philosophy and social science (and the theory of evolution) and see how the method works.
If we have a succession U_1, u_2, u_3, ..., a pair differences of consecutive terms are called first-order differences to the differences of these differences are called second order, etc.. Thus, d_1 (U_1) = u_2-U_1, d_1 (u_2) u_2 = u_3-etc. And d_2 (U_1) = d_1 (u_2)-d_1 (U_1) d_2 (u_2) = d_1 (u_3)-d_1 (u_2), etc.
notation can be daunting, so it is better to see an example:
12 .... 40 .... 90 .... 168 .... 280 ... ....... 432
u_k
........................................ ... 28 ... 112 ..... 50 .... 78 ... 152 ................................. ............. d_1 (u_k)
...... 22 .... 28 ..... 34 ..... 40 ....... .......................................... d_2 (u_k)
......... 6 ...... 6 ...... 6 .......................... ........................... d_3 (u_k)
............. 0 ... ... 0 .............................................. .......... d_4 (u_k)
(I put points for alignment purposes.)
If you look at that u_2 = U_1 + d_1 (U_1) - and, in general, each term in the triangle is equal to its left over which is immediately below: 50 = 28 +22, for example - then achieved a formula for the nth term that follows the law of Newton's binomial coefficients in .
u_n = u_ (n-1) + d_1 (u_ (n-1)) and expands d_1 in terms of the law of formation described above. (Its expansion is left as an exercise.)
As an input (and support) we see the expansion and u_3 u_2:
u_2 = U_1 + d_1 (U_1)
u_3 = u_2 + d_1 (u_2) = [U_1 + d_1 ( U_1)] + [d_1 (U_1) + d_2 (U_1)] = U_1 +2 d_1 (U_1) + d_2 (U_1)
u_4 = u_3 + d_1 (u_3) = [U_1 +2 d_1 (U_1) + d_2 (U_1) ] + d_1 (u_2) + d_2 (u_2) U_1 ................=[
+2 d_1 (U_1) + d_2 (U_1)] +
.... ..................+ d_1 (U_1) + d_2 (U_1) +
................... ............+ d_2 (U_1) + d_3 (U_1)
Therefore: u_4 d_1 +3 = U_1 (U_1) +3 d_2 (U_1) + d_3 (U_1)
can see The coefficients are those of the binomial cube. If the expansion would continue to u_n = U_1 + C (n-1, 1) d_1 (U_1) + C (n-1, 2) d_2 (U_1) + C (n-1, 3) d_3 (U_1) + ...
(C (n-1, k) means "combinations of n-1 'k".) Is shown using mathematical induction. (Also left as an exercise.)
For the above example u_n = 12 +28 (n-1) +1 / 2 [22 (n-1) (n-2)] +1 / 6 [6 (n -1) (n-2) (n-3). That is, u_n = n ^ 3 +5 n ^ 2 +6 n. Sincerely JMD
in VL
note on mathematical notation: the unresolved problem of mathematical notation on the web requires me to use text mode; u_k means or subscript k, n ^ k means to the power n k; ...
